Particle Kinetics Homework

Transcription

1 Chapter 4: article Kinetics Hoework Chapter 4 article Kinetics Hoework Freefor c

2 Chapter 4: article Kinetics Hoework 4-2 Freefor c 2018

3 Chapter 4: article Kinetics Hoework Hoework H.4. Given: article (of ass ) oves along a rough surface (with a coefficient of kinetic friction of µ k between and the surface). t one instant when is at a position on the path that has a radius of curvature of ρ, has a speed of v with the velocity vector for being oriented at a clockwise angle of θ fro the vertical. Find: For this position, (a) Deterine the noral coponent of force of the path acting on particle. (b) Deterine the rate of change of speed of particle. y µ k v θ g T view x HRIZNTL LNE Use the following paraeters in your analysis: µ k = 0.2, = 4 kg, ρ = 0.5, θ = and v = 30 /s. Freefor c

4 Chapter 4: article Kinetics Hoework Hoework H.4. Given: article (of ass ) is attached to an inextensible cable, with the opposite end of the cable being attached to a fixed pin. is also constrained to ove along the rough surface of a right circular cone, with following a circular path of radius r. The coefficient of kinetic friction between and the cone is known to be µ k. The angle of cable with respect to the vertical is known to be θ. t a particular instant in tie, is oving with a speed of v. Find: For this instant in tie, (a) Deterine the noral coponent of force of the cone acting on particle. (b) Deterine the rate of change of speed of particle. L v g θ r r µ k FRNT view T view of the path of Use the following paraeters in your analysis: µ k = 0.3, = 10 kg, θ = 36.87, r = 0.5 and v = 1.5 /s. 4-4 Freefor c 2018

5 Chapter 4: article Kinetics Hoework Hoework H.4.C Given: slotted ar is rotating about end with a constant rate of Ω. article (of ass ) is attached to link, with the ass of assued to be negligible copared to the ass of. is constrained to ove within a sooth, straight slot. t the position of interest, is at an angle of θ easured counterclockwise fro the slotted ar. Note that is a two-force eber. The echanis oves in a horizontal plane. Find: For this position, (a) Deterine the velocity and acceleration of. Write your answers as vectors, in ters of their xy coordinates. It is suggested that you use the oving reference frae equations in your analysis for the velocity and acceleration of. Use an observer attached to the slotted ar. (b) Deterine the noral force acting on and the force acting on by link. y L Ω θ negligible ass x d HRIZNTL plane Use the following paraeters in your analysis: = 15 kg, θ = 36.87, d = 0.5, L = 1.5 and Ω = 4 rad/s. Freefor c

6 Chapter 4: article Kinetics Hoework Hoework H.4.D Given: locks and (having asses of and 2, respectively) are constrained to ove along a sooth inclined surface. Cable (1) is connected to fixed ground at D and to the center of pulley C, as shown, with cable (1) being wrapped around a pulley connected to block. second cable (2) is connected between the fixed ground at D and block. The pulleys are to be assued to be of negligible ass, and the cables are assued to be inextensible and not allowed to go slack. The sections of the cables not wrapped around pulleys are parallel to the incline on which blocks and ove. force F acts along the direction of the incline on block. Find: For this proble, deterine the accelerations of blocks and. F s s sooth θ (1) C s C D (2) 2 g sooth 4-6 Freefor c 2018

7 Chapter 4: article Kinetics Hoework Hoework H.4.E Given: locks and (having asses of and 2, respectively) are constrained to ove along the sooth surfaces shown in the figure below. Meber, in the shape of a quarter-circle arc, connects blocks and, with having a ass that is negligible copared to the asses of and. t the position shown, when the center of the circular arc is directly below block, the syste is released fro rest. Find: For this position, deterine the acceleration of blocks and on release. 2 g negligible ass sooth R sooth HINT: Note that is a two-force eber. What does this say about the direction of the reaction forces on blocks and due to eber? Freefor c

8 Chapter 4: article Kinetics Hoework Hoework H.4.F Given: article (having a ass of ) is constrained to ove around the wall of a horizontal circular cavity, with the path of in the cavity being a circle of radius R. The horizontal surface on which oves is sooth, with the wall of the cavity along which oves is rough having a coefficient of kinetic friction between the wall and of µ k. When at position, is known to have a speed of v. Find: For this proble, (a) Show that the speed of as it oves around the cavity is governed by the differential equation: dv/ds = µ k v/r, where s is the distance traveled by. (b) Using the result of (a) above, deterine the speed v of as a function of s as it oves around the cavity wall. (HINT: Integrate the differential equation found in (a).) Leave your answer in ters of, at ost, v, µ k, R and s. (c) How far does travel before it coes to rest? o/on of in the HRIZNTL plane v R µ k R s sooth T view SIDE view 4-8 Freefor c 2018

9 article Kinetics Hoework robles ME 274 Chapter 4: article Kinetics Hoework roble IV-49 Hoework H.4.G Given: article, having a ass of, is pressed against a spring (having a stiffness Given: articleof, k) having with the a ass spring ofbeing, iscopressed against by an a aount spring (having of!. Upon a stiffness release offro k) with the spring beingrest, copressed the particle by an travels aount along of. a rough Uponhorizontal release fro surface rest, for thewhich particle the travels kinetic along a rough horizontal coefficient surface for of friction which the is known kinetic to coefficient be µ K. of friction is known to be µ k. Find: Find: Deterine Deterine the totalthe distance total distance d traveled d traveled by before by before coing coing to rest. to ssue rest. ssue that d >. Leave your answer that ind sybolic >!. Leave for. your answer in sybolic for. rough, µ K! d Freefor c

10 Chapter 4: article Kinetics Hoework Hoework H.4.H Given: locks and (having asses of 2 and, respectively) are connected by rigid, assless rod of length L. lock is constrained to ove along a sooth vertical guide, and oves along a sooth, horizontal surface. The syste is released fro rest when is at a height of h 1 VE the path of. Hoework robles ME Find: Deterine the speed of block when has dropped to a position that is at a distance of cks and (having asses of 2 and, respectively) are connected by h 2 ELW the path of. id, assless rod of length L. lock is constrained to ove along a ooth vertical guide, and oves ng a sooth, horizontal surface. e syste is released fro rest when 2 s at a height of h 1 VE the path. r this proble, ) Draw a free body diagra of the syste ade up of block, block and rod. g L h 1 ) Deterine the speed of block when has dropped to a position that is at a distance of h 2 ELW the path of. ysis, use the following paraeter 10 kg, k = 1000 N/, L = eters and h 2 = 0.3 eters. Use the following paraeters in your analysis: = 10 kg, L = 0.5, h 1 = 0.4 and h 2 = Freefor c 2018

11 Chapter 4: article Kinetics Hoework Hoework H.4.I Given: article (of ass ) is constrained to ove along a sooth circular guide (of radius R). spring of stiffness k is attached between and the fixed point, where is on the sae horizontal line as the botto of the circular guide, as shown in the figure. t position 1, is at rest and is on the sae horizontal line as the center of the guide C. t position 2, is on the sae horizontal line as and the spring is unstretched. Find: Deterine the speed of particle at position 2. d sooth guide position 1 g R C k position 2 Use the following paraeters in your analysis: = 50 kg, R = 0.6, d = 0.8 and k = 500 N/. Freefor c

12 Chapter 4: article Kinetics Hoework Hoework H.4.J Given: articles and (having asses of and 2, respectively) are attached to a lightweight rigid bar as shown in the figure. constant horizontal force F acts on particle. t the initial state of θ = 0, the syste is at rest. Find: Deterine the angular speed of the bar as a function of θ and in ters of the paraeters of the proble. L / 2 g 2 L / 2! F initial position 4-12 Freefor c 2018

13 Chapter 4: article Kinetics Hoework article Kinetics Hoework robles ME 274 Hoework roble H.4.K IV-14 cannonball of ass is fired toward a steel barrier on a stationary cart. t soe Given: cannonball of ass is fired toward a steel barrier on a stationary cart. t soe tie after rebounding tie after fro rebounding the barrier, fro the the cannonball barrier, the is cannonball observed to is have observed a speed to of have v and a speed is oving of v in the direction and oving shownin below a direction in the figure. shown Let below M in bethe thefigure. cobined Let ass M be of the the cobined cannon/cart. ass ssue of the that the cannon cart isand ablecart. to ove ssue without that the friction cart is along able the to ove horizontal without surface friction andalong ignorethe the influence of air resistance. horizontal surface and ignore the influence of air resistance. Find: For a) thisdeterine proble: the velocity (both agnitude ND direction) of the cart after the (a) Deterinecannonball the velocitybounces vector off the the cart steel after barrier theat cannonball the instant bounces shown below. off the steel barrier at the instant b) Let shown!t represents below; the elapsed tie between the firing of the cannonball and the (a) If t represents instant the shown elapsed below. tiedeterine between the firing average of value the cannonball of the horizontal and theforce instant acting shown below, deterine on the cobined the averagecannon value of and the cart horizontal over the force tie acting period on of the 0 < cobined t <!t. cannon/cart over the tie period of 0 < t < t. v! cannon cart M Use the following paraeters in your analysis: g = 60 lb, Mg = 200 lb, t = 0.2 s, θ = 20 and v = 120 ft/s. Use the following paraeters in your analysis: g = 50 lb, Mg = 250 lb,! t = 0.4 s,! = 25 and v = 100 ft / s. Freefor c

14 Given: lock (of ass 2) is able to slide along a sooth horizontal surface. lock (of ass ) is able to slide along the rough top surface of block, as shown in the figure. Initially, is traveling to the right with a speed of v 1, and block is traveling to the left with a speed of v 1. Chapter 4: article Kinetics Hoework Find: Hoework What H.4.L is the velocity of block when block has to coe rest relative to block? Write your answer as a vector. Use the following paraeters in your Given: lock (of ass 2) is able to slide along a sooth horizontal surface. lock (of ass ) is able analysis: to slide along = 25kg the, rough v 1 = top 26eters surface/ of sec block and, v 1 as = shown 10eters in the/ sec figure.. Initially, is traveling to the right with a speed of v 1, and block is traveling to the left with a speed of v 1. HINT: Consider applying the linear ipulse oentu equation to the syste Find: Deterine ade up of the velocity and together. of block when block has to coe rest relative to block. v 1 rough v 1 2 sooth 4-14 Freefor c 2018

15 article Kinetics Hoework robles ME 274 Chapter 4: article Kinetics Hoework Hoework H.4.M roble IV-47 Given: Find: article (having a ass of ) is constrained to ove within a circular slot (of radius R) that is cut into block (having a ass of M). The syste is released fro rest with particle on a horizontal line passing through the circle s center. Consider all surfaces to be sooth. For this proble: a) Deterine the velocities of and when has oved position 2 Given: article (having awhere ass is ofdirectly ) isbelow constrained. Write your toanswers oveas within vectors. aconsider circular slot (of radius R) that is cut into block (having using athe ass work/energy M). and The linear syste ipulse/oentu is released equations frofor rest the with particle on syste of +. a horizontal line passing through the circle s center. Consider all surfaces to be sooth. b) Deterine the work done on block in oving fro position 1 to position 2. Clearly indicate the sign of your answer (positive or negative). Consider using the work/energy equation for the syste of Find: For this proble: alone. (a) Deterine the velocities of and when has oved position 2 where is directly below (write your Use answers the following as vectors); paraeters in your analysis: = 20kg, M = 40kg and (b) Deterine the R = 0.5 work eters done. on block in oving fro position 1 to position 2. g R!"#$%$"&'('!"#$%$"&')' Use the following paraeters in your analysis: = 20 kg, M = 40 kg and R = 0.5. Freefor c

16 Chapter 4: article Kinetics Hoework Hoework H.4.N Given: locks and C (having asses and 2, respectively) are connected by an inextensible cable. This cable has been fed through a third block (having a ass of 2), as shown in the figure below. The syste is released fro rest with being located at a distance of D below. In the subsequent otion, block contacts block and sticks after a short tie t following their initial contact. ssue that the cable does not break and that the ass of the two pulleys is negligible. Find: For this proble: (a) Deterine the axiu height attained by block after its contact with and sticking to block. (b) Deterine the average tension force in the cable during the during of the ipact tie t between and. g 2 C 2 d Use the following paraeters in your analysis: = 20 kg, d = 2 and t = s Freefor c 2018

17 Chapter 4: article Kinetics Hoework Hoework H.4. Given: Spheres, and C (having asses of 3, 2 and, respectively, are able to slide on a sooth horizontal surface. Spheres and C oving with speeds of v 1 and v C 1, respectively, strike the stationary sphere at exactly the sae tie. The coefficient of restitution is e for all ipacts. Find: Deterine the direction of travel for after the ipact. 2 3 v 1 C v C1 Use the following paraeters in your analysis: = 2 kg, v 1 = 10 /s, v C1 = 4 /s and e = 0.6. Freefor c

18 Chapter 4: article Kinetics Hoework Hoework H.4. Given: articles and ipact each other with initial velocities shown in the figure. coefficient of restitution for this ipact is e. The Find: For this proble: (a) Deterine the velocities of and after the ipact. Write your answers as vectors. (b) Deterine the energy loss during the ipact as a percentage of the initial energy of the syste. 2! v 1 v 1 HRIZNTL LNE Use the following paraeters in your analysis: = 2 kg, v 1 = 10 /s, v 1 = 10 /s, θ = 30 and e = 0. v 1 2 v 1 sooth 4-18 Freefor c 2018

19 Chapter 4: article Kinetics Hoework Hoework H.4.Q Given: Seven particles, through G, ove within a single plane. The ass of each particle is shown in the figure below, along with the velocity and position of each particle. Find: Deterine the total angular oentu about the fixed point for the entire syste of seven particles. y ( 2,2) ft 2 1 ft / s C (2,2) ft G 8 ft / s 8 ft / s ( 2, 1) ft F 10 ft / s 4 ft / s 6 ft / s E 6 ft / s x ( 2, 2) ft D (1.5, 2) ft (2, 2) ft Use the following paraeter in your analysis: = 10 slugs. Freefor c

20 article Kinetics Hoework robles ME 274 roble IV-51 Chapter 4: article Kinetics Hoework Given: particle, having a ass of = 10 kg, is free to slide on a sooth, Hoework H.4.Rlightweight bar. The bar is free to rotate in a horizontal plane about a vertical shaft passing through. spring, having a stiffness k = 400 N/ and Given: particleunstretched, having alength ass of R 0, = 2 iseters, free to is slide connected a sooth, between lightweight and. The bar. spring The bar is free to rotate in is a horizontal copressed plane to half about of its a vertical unstretched shaftlength passing and through released. when spring, the bar having has a stiffness k and unstretched a rotational length speed R 0, of is! connected 1 = 5 rad/sec, between as shown andin. the The figure spring below is copressed left. to half of its unstretched length and released when the bar has a rotational speed of ω 1, as shown in the figure below left. Find: For the position when R = R 0 (shown in the figure below right): Find: For the position a) when Deterine R = Rthe 0 (shown rotation inrate the figure! 2 of the below bar. right): (a) Deterine the rotation rate ω 2 of the bar; b) Deterine the value of dr (b) Deterine the value of Ṙ. dt. R = R 0 /2! 1 k! 2 k R = R 0 HRIZNTL LNE Use the following paraeters in your analysis: = 10 kg, k = 400 N/, R 0 = 2 and ω 1 = 5 rad/s Freefor c 2018

21 Chapter 4: article Kinetics Hoework Hoework H.4.S Given: article (having a ass of ) is attached to rod, with end being pinned to a vertical shaft. article is also attached to particle (having a ass of ) through rod, with particle being constrained to slide along the rotation axis of the shaft. t a tie when the shaft is rotating with a rate of ω 1 and with θ = θ 1, and while particles and are stationary with respect to the shaft, the particles are released. ssue all surfaces to be sooth and the ass of rods and to be negligible. Find: Deterine the speed of particle when θ = θ 2. sooth bearings L g θ L ω sooth bearings Use the following paraeters in your analysis: = 20 kg, L = 0.5, ω 1 = 6 rad/s, θ 1 = 30 and θ 2 = 90. Freefor c

22 Chapter 4: article Kinetics Hoework Hoework H.4.T ics Hoework robles ME 274 Given: Rigid ar (having length L and having negligible ass) is pinned to ground at end. particle of ass M is attached to end of. t instant 1, a pellet (having a ass of ) strikes the stationary particle with a speed of v 1 in the direction shown below in the figure. IV-19 t the end of a short tie interval ipact, sticks to. (having length L and having negligible ass) is pinned to ground at end. of ass M is attached Find: Deterine to end of the. angular t instant speed 1 of ar a pellet iediately (having a ass after sticks to. es the stationary particle with a speed of v 1 in the direction shown below in t the end of a short tie interval ticks to. w a free body diagra (FD) of the te ade up of particles and during act. Recall that and are particles., the physical diensions of these two ies are negligible). ased on your FD, vide arguent that supports the clai t the angular oentu for syste bout is conserved during ipact. terine the angular speed of ar iediately after sticks to. nsider using the angular oentu ation. g! L M v 1 VERTICL LNE Use the following paraeters in your analysis: φ = 30, L = 4 ft, g = 4 lb, Mg = 8 lb and llowing paraeters v 1 = in 150 your ft/s. analysis:! = 36.87, L = 5 ft, g = 2 lb, and v 1 = 120 ft / s Freefor c 2018